Gevrey-modulation spaces and smoothing effect for the nonlinear Schrödinger equations

Abstract

We study the global Cauchy problem for the nonlinear Schrödinger equations with the power type nonlinearity or the Hartree type nonlinearity, in the mass critical setting. Especially, we show the Gevrey smoothing effect for the nonlinear Schrödinger equations with data which satisfy sub-exponentially decaying condition and has sufficiently small norm. Also we show the existence of scattering state in the class of sub-exponentially decaying functions without loss of radius of convergence for sufficiently small data.